Equilibria and Kinetics ofBiological Macromolecules

Equilibria and Kinetics ofBiological Macromolecules

Jan HermansBarry Lentz

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Hermans, Jan.Equilibria and kinetics of biological macromolecules / by Jan Hermans and Barry Lentz.

p. ; cm.Includes index.ISBN 978-1-118-47970-4 (cloth)I. Lentz, Barry. II. Title.[DNLM: 1. Biocompatible Materials–pharmacokinetics. 2. Biophysical Processes. 3.

Macromolecular Substances–pharmacokinetics. 4. Molecular Conformation. QT 37]RM301.5615.7–dc23

2013013996

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

For Harold Scheraga

Contents

Preface xix

Acknowledgments xxi

PART 1 BASIC PRINCIPLES 1

1. Thermodynamics 3

1.1 Introduction / 3

1.2 The fundamental postulates or Laws of thermodynamics / 4

1.3 Other useful quantities and concepts / 14

1.4 Thermodynamics of the ideal gas / 19

1.5 Thermodynamics of solutions / 20

1.6 Phase equilibria / 25

1.7 Chemical equilibria / 29

1.8 Temperature dependence of chemical equilibria: The van’t Hoffequation / 31

1.9 Microcalorimetry / 31

Notes / 33

vii

viii CONTENTS

2. Four Basic Quantum Mechanical Models of Nuclear and ElectronicMotion: A Synopsis 35

2.1 Introduction / 35

2.2 Fundamental hypotheses of quantum theory / 36

2.3 Three simple models of nuclear motion / 38

2.4 Hydrogen atomic orbitals: A simple model of electronic motionin atoms / 44

2.5 Many electron atoms / 47

Notes / 49

Suggested reading / 49

3. Molecular Structure and Interactions 51

3.1 Introduction / 51

3.2 Chemical bonding: Electronic structure of molecules / 51

3.3 Empirical classical energy expressions / 58

3.4 Noncovalent forces between atoms and molecules / 62

3.5 Molecular mechanics / 70

Notes / 75

Suggested reading / 76

4. Water and the Hydrophobic Effect 77

4.1 Introduction / 77

4.2 Structure of liquid water / 78

4.3 The hydrophobic effect / 84

Notes / 89

Suggested reading / 89

PART 2 STATISTICAL MECHANICS: THE MOLECULAR BASISOF THERMODYNAMICS 91

5. The Molecular Partition Function 93

5.1 Introduction / 93

5.2 The Maxwell–Boltzmann distribution / 93

CONTENTS ix

5.3 The molecular partition function and thermodynamicfunctions / 99

5.4 Application to macromolecules / 101

Notes / 108

Suggested reading / 110

6. System Ensembles and Partition Functions 111

6.1 Introduction / 111

6.2 Closed systems: The canonical ensemble / 112

6.3 The canonical partition function of systems with continuousenergy distributions: The phase-space integral / 119

6.4 Application: Relation between binding and molecular interactionenergy / 123

6.5 Application: Binding of ligand to a macromolecule / 125

6.6 Open systems: The grand canonical ensemble or grandensemble / 127

6.7 Fluctuations / 131

6.8 Application: Light scattering as a measure of fluctuationsof concentration / 134

Notes / 135

Suggested reading / 136

7. Sampling Molecular Systems with Simulations 137

7.1 Introduction / 137

7.2 Background / 138

7.3 Molecular dynamics / 139

7.4 Metropolis Monte Carlo / 142

7.5 Simulation of a condensed system / 143

7.6 Connecting microscopic and macroscopic system properties / 144

7.7 An example: Dynamics of Ace-Ala-Nme in solution / 146

7.8 Forced transitions / 149

7.9 Potential of mean force for changes of chemistry: “ComputerAlchemy” / 152

x CONTENTS

7.10 The potential of mean force and the association equilibriumconstant of methane / 157

Notes / 158

Suggested reading / 159

PART 3 BINDING TO MACROMOLECULES 161

8. Binding Equilibria 163

8.1 Introduction / 163

8.2 Single-site model / 163

8.3 Measuring ligand activity and saturation / 166

8.4 Multiple sites for a single ligand / 173

8.5 A few practical recommendations / 182

Notes / 183

Suggested reading / 184

9. Thermodynamics of Molecular Interactions 185

9.1 Introduction / 185

9.2 Relation between binding and chemical potential: Unifiedformulation of binding and “exclusion” / 186

9.3 Free energy of binding / 187

9.4 Mutual response / 188

9.5 Volume exclusion / 189

9.6 Accounting for interactions of macromolecule and solventcomponents / 193

Notes / 196

Suggested reading / 196

10. Elements of Statistical Mechanics of Liquidsand Solutions 197

10.1 Introduction / 197

10.2 Partition function of ideal solution from thermodynamics / 198

10.3 Statistical mechanics of the ideal solution / 200

CONTENTS xi

10.4 Formulation of molecular binding interactions in terms of apartition function: Empirical approach based onthermodynamics / 202

10.5 A purely statistical mechanical formulation of molecular bindinginteractions / 204

10.6 Statistical mechanical models of nonideal solutionsand liquids / 208

Notes / 211

Suggested reading / 211

11. Analysis of Binding Equilibria in Terms of PartitionFunctions 213

11.1 Alternate equivalent representations of the partition function / 213

11.2 General implications / 215

11.3 Site-specific binding: General formulation / 216

11.4 Use of single-site binding constants / 218

11.5 Partition function for site binding: One type of ligand,independent multiple sites / 220

11.6 Site binding to interdependent or coupled sites / 221

Suggested reading / 222

12. Coupled Equilibria 223

12.1 Introduction / 223

12.2 Simple case: Coupling of binding (one site) and conformationchange / 224

12.3 Coupling of binding to multiple sites and conformationchange / 225

12.4 Free energy of binding can “drive” conformation change / 230

12.5 Formation of oligomers and polymers / 232

12.6 Coupled polymerization and ligand binding / 237

Notes / 238

Suggested reading / 238

13. Allosteric Function 239

13.1 Introduction / 239

xii CONTENTS

13.2 Background on hemoglobin / 240

13.3 The allosteric or induced-fit model of hemoglobin / 241

13.4 Simplified allosteric models: Concerted and sequential / 242

13.5 Numeric example / 244

13.6 Comparison of oxygen binding curves / 245

13.7 Separating oxygen binding and conformation change ofhemoglobin / 246

13.8 Experiments with hybrid hemoglobins / 248

13.9 Two-site proteins, half-the-sites reactivity, and negativecooperativity / 248

13.10 Allosteric effects in protein function / 249

13.11 Sickle cell hemoglobin / 250

13.12 Hill plot / 250

Notes / 252

Suggested reading / 253

14. Charged Groups: Binding of Hydrogen Ions, Solvation,and Charge–Charge Interactions 255

14.1 Introduction / 255

14.2 Ionizable groups in peptides / 256

14.3 pH titration of a protein: Ribonuclease—normal and abnormalionizable groups / 257

14.4 Local interactions cause pKas to be abnormal / 260

14.5 Internal charge–charge interactions: Ion pairs or salt bridges / 260

14.6 Measuring stability of salt bridges from double mutant cycles / 261

14.7 Salt bridges stabilize proteins from thermophilic organisms / 262

14.8 Charged side chains in enzyme catalysis and protein solubility / 263

14.9 Accounting for charge–charge and charge–solvent interactions / 263

14.10 The continuum dielectric model / 264

14.11 Application to a charged spherical particle / 266

14.12 Accounting for ionic strength: The Poisson–Boltzmann equationand Debye–Huckel theory / 267

CONTENTS xiii

14.13 Numerical treatment via finite differences / 268

14.14 Strengths and limitations of the continuum dielectric model / 269

14.15 Applications of the continuum dielectric model tomacromolecules / 270

Notes / 273

Suggested reading / 275

PART 4 CONFORMATIONAL STABILITYAND CONFORMATION CHANGE 277

15. Some Elements of Polymer Physics 279

15.1 Introduction / 279

15.2 Conformational variation in small molecules / 280

15.3 Conformational variation in chain molecules / 280

15.4 The ideal random coil and the characteristic ratio / 281

15.5 The persistence length as a measure of chain flexibility / 282

15.6 Conformation of self-avoiding chains / 283

15.7 Dependence of chain conformation on solvent conditions; “Theta”conditions / 284

15.8 Relating chain statistics to molecular structure / 286

15.9 Polyelectrolytes / 287

Notes / 288

Suggested reading / 289

16. Helix-Coil Equilibria 291

16.1 Introduction: Multistate transitions of helical polymers / 291

16.2 Single-stranded poly (A): A completely non-cooperativetransition / 291

16.3 Synthetic polypeptides / 292

16.4 Zimm–Bragg, Gibbs–DiMarzio, and Lifson–Roig analyses / 295

16.5 Solution of the partition function / 297

16.6 Experiments on synthetic homo-polypeptides and proteinfragments / 299

xiv CONTENTS

16.7 Experimental determination of helix propensities in syntheticpeptides / 299

16.8 Helix stabilization by salt bridges in oligomers containingGlu and Lys / 301

16.9 Helix stabilization by charged groups interacting with the helixdipole / 303

16.10 Helix-coil equilibria of nucleic acids / 303

16.11 Melting transition of DNA / 306

Notes / 309

17. Protein Unfolding Equilibria 311

17.1 Introduction / 311

17.2 The two-state approximation / 312

17.3 Working with the two-state model / 314

17.4 Calorimetric measurements of the thermodynamics of proteinunfolding / 316

17.5 Unfolding thermodynamics of ribonuclease / 318

17.6 Cold denaturation / 322

17.7 Solvent-induced unfolding: Guanidine hydrochloride and urea / 322

17.8 Mixed solvents: Denaturants and stabilizers / 324

17.9 Unfolding is not two-state under native conditions: Hydrogenexchange / 328

17.10 Nature of the two states / 332

17.11 A third state: The molten globule / 336

17.12 Range of stability / 338

17.13 Decomposition of the thermodynamic parameters forunfolding / 340

Notes / 342

Suggested reading / 345

18. Elasticity of Biological Materials 347

18.1 Background / 347

18.2 Rubber-like elasticity of polymer networks / 348

CONTENTS xv

18.3 Theory of rubber elasticity / 349

18.4 Rubber-like elasticity of elastin / 351

18.5 Titin and tenascin: Elasticity based on transitions betweenconformation states / 352

18.6 Single-molecule force-extension measurement / 354

Notes / 355

PART 5 KINETICS AND IRREVERSIBLE PROCESSES 357

19. Kinetics 359

19.1 Introduction / 359

19.2 Measuring fast kinetics by rapid perturbation / 360

19.3 Fast rates from spectroscopic line shape and relaxation times / 362

19.4 Relaxation time in a unimolecular reaction / 364

19.5 Relaxation time in a bimolecular reaction / 365

19.6 Multiple reactions / 367

19.7 Numeric integration of the master equation / 367

19.8 Consecutive reactions cause delays / 368

19.9 Steady state assumption: Michaelis–Menten model, microscopicreversibility, and cyclic processes / 369

19.10 Nucleation and growth mechanisms in phase transitions andbiopolymer folding reactions / 372

19.11 Kinetic theory and the transition state / 373

19.12 Transition state in catalysis / 375

19.13 Inhibitor design: Transition state analogs / 377

19.14 The diffusion-limited reaction / 379

19.15 Estimating reaction rates from simulations / 381

Notes / 386

Suggested reading / 387

20. Kinetics of Protein Folding 389

20.1 Introduction / 389

xvi CONTENTS

20.2 Slow folding: Misfolding / 390

20.3 Slow folding: Cis–trans isomerization of proline / 391

20.4 Slow folding: Disulfide bond formation / 392

20.5 Two-state folding kinetics / 393

20.6 Folding rates of some peptides and proteins / 395

20.7 Probing the transition state: Tanford’s β value and Fersht’s φvalue / 398

20.8 Early events in folding / 400

20.9 (Free) energy landscape for folding / 402

20.10 The “Levinthal Paradox” and the folding funnel / 403

20.11 Transition state(s), pathway(s), reaction coordinate(s) / 404

20.12 Computer simulations of protein folding and unfolding / 405

20.13 Conclusion / 410

Notes / 410

Suggested reading / 412

General references / 413

21. Irreversible and Stochastic Processes 415

21.1 Introduction / 415

21.2 Macroscopic treatment of diffusion / 416

21.3 Friction force opposes motion / 417

21.4 Random walk as a model diffusive process / 418

21.5 Equation of motion for stochastic processes: The Langevinequation / 419

21.6 Fluctuation–dissipation theorem / 420

21.7 Specific examples of fluctuating force / 421

21.8 Alternative form of the fluctuation–dissipation theorem / 422

21.9 Diffusive motion and the Langevin equation / 424

21.10 Smoluchowski and Fokker–Planck equations / 425

21.11 Transition state theory revisited / 429

21.12 Kramers’ theory of reaction rates / 432

CONTENTS xvii

Notes / 435

Suggested reading / 436

APPENDICES 437

A. Probability 439

A.1 Introduction / 439

A.2 Sample statistics / 440

A.3 Probability distributions / 440

A.4 A few comments / 442

A.5 Fitting theory to data: Computer-facilitated “Least Squares” / 442

B. Random Walk and Central Limit Theorem 445

B.1 Introduction / 445

B.2 Random selection / 445

B.3 The central limit theorem / 446

B.4 Simple random walk / 447

C. The Grand Partition Function: Derivation and Relation to OtherTypes of Partition Functions 449

C.1 Introduction / 449

C.2 Derivation / 450

C.3 Connection with thermodynamic functions / 451

C.4 Relation to other types of partition functions / 453

D. Methods to Compute a Potential of Mean Force 457

D.1 Introduction / 457

D.2 Thermodynamic integration / 458

D.3 Slow growth / 458

D.4 Thermodynamic perturbation / 459

D.5 Umbrella sampling / 460

D.6 Conclusion / 461

xviii CONTENTS

E. Theory of the Helix-Coil Transition 463

E.1 Introduction / 463

E.2 Maximum term solution / 464

E.3 Solution via matrix algebra / 466

F. Laplace Transform 469

F.1 Solving linear differential equations with the Laplacetransform / 469

F.2 The Laplace transform / 469

F.3 Two key properties of the Laplace transform / 470

F.4 Example 1: The Poisson process (or consecutive reactions) / 471

F.5 Example 2: General case of linear kinetic equations / 472

F.6 Example 3: Coupled harmonic oscillators—normal modes / 474

F.7 Table of inverse Laplace transforms / 476

G. Poisson Equation 477

G.1 Formulation / 477

G.2 Exact solution for a simple case: The Born model / 478

G.3 Accounting for ionic strength: Poisson–Boltzmann equationand Debye–Huckel theory / 480

H. Defining Molecular Boundaries 483

I. Equations 485

I.1 Stirling’s formula and combinatorials / 485

I.2 Integrals of Gaussian distributions / 486

I.3 Cartesian and spherical polar coordinates / 486

I.4 Laplace operator in three-dimensional cartesian, polar,and cylindrical coordinates / 487

I.5 Sums of geometric series / 487

I.6 The Kronecker and Dirac delta functions / 488

I.7 Useful relations between differential quotients / 488

I.8 Random numbers / 489

Index 491

PrefaceIt is only by attempting to explain our science to each other that

we find out what we really know.—J. M. Ziman, Nature 252: 318–324 (1969)

This book has grown out of circa 12 years of collaborative teaching of a 6-creditbiophysics course that forms the core of the didactic teaching for the Molecular andCellular Biophysics Program at UNC CH. Thus the book is directed at an audienceof first year graduate students. However, the book has grown well beyond thecontent of those courses, also thanks to input and suggestions from colleagues whohave shared our teaching the course (see Acknowledgments), and it is our hopethat it will prove useful to working biochemists who seek a deeper understandingof modern biophysics.

The book is not meant to be a complete text in biophysics, as it focuses onthe input of physics and physical chemistry to experimental studies and theoreticalmodels of equilibria and kinetics of biological macromolecules (largely, proteins).A chapter is devoted to methods of molecular simulations; applications of moleculardynamics are included in several chapters. On the other hand, we limited the sizeof this book by devoting no space to spectroscopy and structure determination.

The book assumes some knowledge of physics and/or physical chemistry, butin Part 1, the chapters on thermodynamics, simple quantum mechanics and molec-ular structure and intra- and intermolecular forces shore up what may be shakybackgrounds of some students, and provide references for later chapters. Part 1concludes with a chapter on water and the hydrophobic effect.

Two chapters in Part 2 introduce various ensembles of statistical mechanics, andthese are followed by the aforementioned chapter on molecular simulations.

Next, in Part 3, we discuss equilibria of binding of “ligands” to macromoleculesfrom different standpoints: chemical equilibrium theory, thermodynamics, and sta-tistical mechanics. These are followed by a discussion of linked equilibria, and achapter that focuses on hemoglobin as an example of allosteric control of function.

xix

xx PREFACE

Part 3 concludes with a chapter on charge–charge interactions of macromoleculesin solution.

In Part 4, we deal with folding equilibria. A brief overview of the physics ofpolymer solutions is followed by a chapter on the theory of helix-coil transitionsof polypeptides and its many applications, and it ends with a section on helix-coilequilibria of double-stranded nucleic acids. This is followed by a long chapter onequilibria of protein folding. Part 4 concludes with a chapter on elasticity withelastin and tenascin as examples of two different mechanisms.

The final part of the book is devoted to kinetics. The first chapter describeskinetic measurement methods and a variety of kinetic models, ranging from simplerate equations to transition state theory. This is followed by a chapter on experi-ments and theory of kinetics of protein folding. Part 5 concludes with a chapter onstochastic processes and theories from the Langevin equation to Kramers’ theoryof reaction rates.

Finally, in a series of Appendices we have covered technical (mostly mathemat-ical) details which we had skipped earlier to make the main content of this bookeasier to follow.

The authors will maintain a web page devoted to corrections and discussionof this book. Please consult the authors’ personal web pages at the University ofNorth Carolina.

Acknowledgments

This book’s inception was in the form of lecture notes for the introductory classin molecular biophysics given at UNC each fall semester. An enormous help hasbeen the feedback we received from students taking the class.

We have received help from many colleagues. We are grateful to professorsPapoian (now at the University of Maryland) and Dokholyan, who have each taughtpart of the course, for letting us base important sections of the book on new pre-sentations given by them in their lectures. Individual chapters have had input fromGary Ackers at Washington University, from Gary Pielak in the UNC ChemistryDepartment, from Gary Felsenfeld at the NIH, from Andy McCammon at UCSD,from Weitao Yang at Duke, from Austen Riggs at the University of Texas, fromRobert Baldwin at Stanford and from Hao Hu at the University of Hong Kong.

We thank Dr. M. Hanrath, University of Cologne for the computer drawingsof hydrogen atom wave functions shown in Chapter 2, and Dr. Chad Petit formicrocalorimeter results discussed in Chapter 8. Some figures of molecular struc-tures were prepared with the vmd graphics program.* We acknowledge manyanswers to questions involving basic Physics, found by consulting Wikipedia.

JH and BRL

September 2012

*Humphrey, W., Dalke, A., Schulten, K. VMD - visual molecular dynamics. J. Mol. Graphics Modell.14: 33–38 (1996).

xxi

Part 1

Basic Principles

In our treatment of the biophysics of macromolecules, we must assume someknowledge of basic physics and physical chemistry. Part 1 is a compendium (“re-view” if you like) of those aspects of these subjects that the reader will be expectedto have mastered. This is not meant to take the place of a thorough textbook dealingwith these topics (some recommended texts are listed), but rather as a summary ofkey concepts and information.

We start with thermodynamics, which is unique as it expressly assumes no mod-els of molecular structure and intermolecular interactions, while this is otherwise thecase for all other topics treated in this book. As thermodynamics remains an essen-tial tool of modern molecular physics, one simply must know thermodynamics, sowe start with it.

We then attend to three basic motions of massive particles such as nuclei oreven whole molecules: free translation, free rotation, and movement in a potential.For simplicity, we limit ourselves to the quantum mechanical treatment of each,although the reader will surely recall classical treatments that occupied parts ofbasic physics courses, which describe the high temperature limits of these motions.The motion of electrons, however, can be described only via quantum mechanics.So, the fourth essential model we review is that of the Hydrogen Atom. Thismasterpiece of late nineteenth and early twentieth century physics provides thebasis for all of chemistry and for what we think we know about molecules and theirinteractions, which is the topic of the third chapter of this section. After reviewing

Equilibria and Kinetics of Biological Macromolecules, First Edition. Jan Hermans and Barry Lentz.© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

1

2 BASIC PRINCIPLES

how quantum methods are used to calculate molecular structure and energies, andthe difficulties of doing these calculations on the grand scale required by studiesof macromolecules, we introduce the approximation of molecular mechanics.

Finally, in Chapter 4 this basic material is applied to the structure of liquid waterand the thermodynamics of the hydrophobic effect.

1Thermodynamics

One does not understand thermodynamics, one can only know it—Jan Hermans

As a biophysicist, you must know thermodynamics—Barry Lentz

1.1 INTRODUCTION

Thermodynamics describes the relation between different forms of energy, theirinterconversion, and the exchange of energy between physical systems. Thermo-dynamics is applicable to energy management in all situations. It was developedin the context of the industrial revolution, with an important goal being the designof more efficient versions of newly invented machines, first the steam engine, latersuch devices as the internal combustion engine and the refrigerator. Thermodynam-ics also describes how the total energy of a system is partitioned between usefulenergy (available to do work) and wasted energy (that associated with the random-ness of a system), and establishes conditions that must be met for a system tonot undergo spontaneous change, that is, to be at equilibrium. The branch of ther-modynamics that concerns us most deals with the energetics of chemical systemsand systems containing interacting molecules. However, thermodynamics does notformally assume a molecular nature of matter, but is simply a formal description ofthe relationship between work, heat, and energy. Three laws, which are based on“everyday” observations, form the foundation of thermodynamics. The surprisinglyprofound conclusions that follow from these laws have been verified extensively.

Thermodynamics strikes many as a boring formalism, seemingly devoid of theinteresting intellectual content of quantum and statistical mechanics. Indeed, one

Equilibria and Kinetics of Biological Macromolecules, First Edition. Jan Hermans and Barry Lentz.© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

3

4 THERMODYNAMICS

can think of thermodynamics as a bookkeeping tool that tracks otherwise obscurerelations between different forms of energy storage, and in doing so keeps thebiophysicist from many an egregious error. At the same time, the very fact that acomplex framework of relations can be built on a few fundamental laws should be asource of marvel, as is the insight of the scientists who developed thermodynamicsin the nineteenth century. The development of thermodynamics on the basis ofa few laws resembles the development of mathematics from a small number ofaxioms. However, the axioms of mathematics can be chosen by the mathematician,while the laws of thermodynamics are based on observations of our physical world,and these laws could be changed only on the basis of radically new experimentalfindings.

This chapter is not a textbook on thermodynamics; it is presumed that studentsusing this book have had an introductory physical chemistry course that treatedchemical thermodynamics in some detail. It is also presumed that many who havehad such a course do not remember it very well. Thus, our goal is to review brieflythe fundamental concepts of thermodynamics and then to give them a context interms of solutions of macromolecules and their interactions with other molecules.

1.2 THE FUNDAMENTAL POSTULATES OR LAWSOF THERMODYNAMICS

1.2.1 Systems

A system is a part of the universe in which we have interest for a particularproblem. In biology, it is often some collection of molecules. It is separated bysome boundary from the rest of the universe (its surroundings; Fig. 1.1).

Open systems exchange energy and matter with their surroundings.

Closed systems exchange energy but not matter with their surroundings.

Isolated systems exchange neither energy nor matter with their surroundings.

System

q = Heat flow into systemfrom surroundings

w = Work done bysystem on

surroundings

Surroundings

FIGURE 1.1 A closed system exchanges energy in the form of heat and work but notmatter with its surroundings. If no heat is exchanged (q = 0), the process is adiabatic. Anopen system can also exchange matter with the surroundings.

THE FUNDAMENTAL POSTULATES OR LAWS OF THERMODYNAMICS 5

1.2.2 States and State Functions

The state of a closed system can be changed by the exchange of energy withthe outside (surroundings), and can also change spontaneously. Thermodynamics isconcerned with the equilibrium states that are the outcome of spontaneous change,and with the processes by which change from one equilibrium state to anotheroccurs. Many equilibrium states are metastable; for example, a mixture of oxygenand hydrogen gases is stable, but can be ignited to explode (spontaneous change)and form water vapor. A state is defined in terms of characteristic properties, suchas temperature, density, pressure, and chemical composition. The energy of a stateis one of its fundamental characteristics, and is therefore called a state function.By definition, a state function depends on certain properties of a system such asthe number of molecules composing it (N ), the volume (V ), perhaps the pressureon the system, and a very interesting property called temperature (T ).

Observation tells us that not all these properties are independent; that is, if weset the values of some, then others are fixed by these assignments. Aside from theextensive (how big the system is) property N , the thermodynamics of a closed sys-tem are defined by two additional properties, which are referred to as independentvariables of the system. All other properties of the system, including its state func-tions, are dependent properties of the system. There is nothing holy or sacrosanctabout an independent variable, these are defined by the experimental conditions weuse to observe the system and are those properties over which we exercise control.However, once we choose these independent properties or variables, the valuesof the state functions for the system are defined and can be obtained by the lawsof thermodynamics. Thermodynamic state functions depend only on the values ofthese independent properties and not on how the system reached this state.

1.2.3 The First Law and Forms of Energy—Energy a State Function

Classical mechanics introduces three forms of energy: kinetic energy, potentialenergy, and work. Kinetic energy is evident in an object’s motion. The poten-tial energy of an object is latent energy that allows the object to do work or toacquire kinetic energy. Work has associated with it a force and a path; force act-ing along the path changes the potential energy and/or the kinetic energy of anobject. Thermodynamics considers an additional category of energy, heat, and isconcerned solely with the relationships between and interconversion of heat, work,and energy. We stress that thermodynamics does not distinguish between kineticand potential energy, nor does it bother itself with motion—these issues are totallythe venue of mechanics. These two independent areas of physics came togetheronly in the latter half of the nineteenth century through the collaboration of theScottish mathematician James Scott Maxwell (kinetic theory of gases) and the Aus-trian physicist Ludwig Eduard Boltzmann (the Boltzmann distribution) to developa statistical description of the average speed of molecules in a gas. This is theMaxwell–Boltzmann distribution, which forms the basis of statistical mechanics(also called statistical thermodynamics; see Chapter 5).

6 THERMODYNAMICS

The First Law of thermodynamics states that the energy of a system and itssurroundings is conserved. The “everyday” experience of doing work to move amass up a hill against the force of gravity leads to the concept that the workdone is converted into potential energy, which remains hidden, until the objectis released and rolls back down. In the absence of friction, energy is conservedduring the rollback down the hill, and the object acquires a new form of energy,kinetic energy. There are many familiar examples of converting energy into work orheat or capturing work as energy. Several are illustrated in Fig. 1.2. The inventionof the steam engine stimulated the development of thermodynamics. In it and itsmodern-day replacement, the internal combustion energy, the energy released inthe form of heat produced when hydrocarbons react with oxygen to form CO2 andH2O, causes this gas mixture (or water vapor in the steam engine) to expand andthis produces pressure–volume work (PV work) on a piston that is captured asthe work needed to increase the kinetic energy of a vehicle. Thus, by virtue ofthe First Law, heat also must be considered a form of energy. In this example,a chemical reaction liberates energy in the form of heat. By virtue of the FirstLaw, the chemical (or internal) energy of the reactants must decrease by a likeamount. Similarly, a charged battery possesses potential energy that is releasedwhen electrons are allowed to flow through a wire to drive an electric motor that

1. Chemical energy → Heat(CH2)n + 3nO2 → n(CO2 + H2O) + q

2. Hot CO2 + H2O expand: q → PV work

3. Car moves:PV work → Kinetic energy

4. Friction:Kinetic energy → Heat

2e−

Zn

Zn++

Cu

Cu++

q

Electrical work: −I2Rt

Electrical

Radiantenergy

Chemical energy → Electrical work

(a) (b)

(c)

(d)

FIGURE 1.2 Examples of interconversion of different forms of energy. (a) Internal com-bustion engine, (b) light bulb, (c) electric water heater, and (d) flashlight battery.